Question

Express the Vigenere cipher as a cryptosystem.

Short answer

$$\begin{gathered} \mathbf{(}\mathbf{P}\mathbf{,}\mathbf{C}\mathbf{,}\mathbf{K}\mathbf{,}\mathbf{E}\mathbf{,}\mathbf{D}\mathbf{)} \\ \mathbf{P}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{C}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{K}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{E}\mathbf{=}\mathbf{\left\{}{\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \\ \mathbf{D}\mathbf{=}\mathbf{\left\{}{\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\mid }\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \end{gathered}$$

Step-by-step solution

Step 1

A CRYPTOSYSTEM is a five- tuple $\mathbf{(}\mathbf{P}\mathbf{,}\mathbf{C}\mathbf{,}\mathbf{K}\mathbf{,}\mathbf{E}\mathbf{,}\mathbf{D}\mathbf{)}$ whereP is the set of plaintext strings, C is the set of cipher text strings, K is the key space, E is the set of encryption functions, and is the set of decryption functions.

The key is a string of letters with numerical equivalents ${\mathbf{k}}_{\mathbf{1}}{\mathbf{k}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{k}}_{\mathbf{m}}$where ${\mathbf{k}}_i\in {\mathit{z}}_{\mathbf{26}}$which implies that the key space k contains all strings of elements in localid="1668669784281" ${\mathbf{Z}}_{\mathbf{26}}$, which can be denoted as$\mathbf{Z}{\mathbf{*}}_{\mathbf{26}}$

_{$\mathbf{K}\mathbf{=}\mathbf{Z}{\mathbf{*}}_{\mathbf{26}}$}

Similarly, the plaintext strings (blocks) and the cipher text strings(blocks) have numerical equivalent with digits in ${\mathbf{Z}}_{\mathbf{26}}$, which implies that the plaintext strings and cipher text strings are also strings of elements in ${\mathbf{Z}}_{\mathbf{26}}$and this again gets denoted by$\mathbf{Z}{\mathbf{*}}_{\mathbf{26}}$

_{$$\begin{gathered} \mathbf{P}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{C}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \end{gathered}$$}

The encryption increases the ${\mathbf{p}}_{\mathbf{i}}$th digit ${\mathbf{k}}_{\mathbf{i}}$ by modulo 26 and thus the set of encryption functions are_{${\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26}$} for k an integer from 0 to $\mathbf{25}\mathbf{(}\mathbf{k}\mathbf{\in }{\mathbf{z}}_{\mathbf{26}}\mathbf{)}$

_{$$\begin{gathered} \mathbf{E}\mathbf{=}\mathbf{\left\{}{\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \end{gathered}$$}

The corresponding decryption then decrease the ${\mathbf{c}}_{\mathbf{i}}$th digit by ${\mathbf{k}}_{\mathbf{i}}$ modulo26 and the set of decryption functions are ${\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26}$ for k an integer from0 to$\mathbf{25}\left(k\mathrm{\in }{z}_{26}\right)$

_{$$\begin{gathered} \mathbf{D}\mathbf{=}\mathbf{\left\{}{\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\mid }\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \end{gathered}$$}

Step 2

We can then describe the Vigen'ere cipher as the following cryptosystem

$$\begin{gathered} \mathbf{(}\mathbf{P}\mathbf{,}\mathbf{C}\mathbf{,}\mathbf{K}\mathbf{,}\mathbf{E}\mathbf{,}\mathbf{D}\mathbf{)} \\ \mathbf{P}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{C}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{K}\mathbf{=}{{\mathbf{Z}}^{\mathbf{\ast }}}_{\mathbf{26}} \\ \mathbf{E}\mathbf{=}\mathbf{\left\{}{\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{E}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{+}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \\ \mathbf{D}\mathbf{=}\mathbf{\left\{}{\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\mid }\mathbf{k}\mathbf{\in }{\mathbf{Z}}_{\mathbf{26}}\mathbf{\right\}} \\ {\mathbf{D}}_{\mathbf{k}}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{mod}\mathbf{26} \end{gathered}$$